Non-local boundary energy forms for quasidiscs: Codimension gap and approximation

TL;DR

This paper studies non-local fractional Laplace energy forms on quasicircles and demonstrates their approximation by polygonal curves using generalized Mosco convergence, addressing dimension-related fractional order mismatches.

Contribution

It introduces a method to approximate non-local boundary energy forms on quasidiscs by polygonal domains, accounting for Hausdorff dimension differences.

Findings

Established convergence of energy forms on quasidiscs to polygonal approximations.

Provided a framework for handling fractional order mismatches due to Hausdorff dimension.

Demonstrated approximation techniques for non-local boundary energy forms.

Abstract

We consider non-local energy forms of fractional Laplace type on quasicircles and prove that they can be approximated by similar energy forms on polygonal curves. The approximation is in terms of generalized Mosco convergence along a sequence of varying Hilbert spaces. The domains of the energy forms are the natural trace spaces, and we focus on the case of quasicircles of Hausdorff dimension greater than one. The jump in Hausdorff dimension results in a mismatch of fractional orders, which we compensate by a suitable choice of kernels. We provide approximations of quasidiscs by polygonal $(\varepsilon,\infty)$-domains with common parameter $\varepsilon>0$ and show convergence results for superpositions of Dirichlet integrals and non-local boundary energy forms.